Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable $X$. Is it true that $$E(X)\leq \liminf_{n\to+\infty}E(X_n)$$
Thanks a lot for the help in advance!
Best Answer
Yes, it's true. By the very definition of $\liminf$, there exists a subsequence such that
$$\liminf_{n \to \infty} \mathbb{E}(X_n ) = \lim_{k \to \infty} \mathbb{E}(X_{n_k}). \tag{1}$$
Since, by assumption, $X_{n} \to X$ in probability (hence in particular $X_{n_k} \to X$ in probability), we can choose a further subsequence of $(X_{n_k})_{k \in \mathbb{N}}$ which converges almost surely to $X$. Now the claim follows directly from Fatou's lemma and $(1)$.